Coordinate Geometry Distance & Midpoint Cheat Sheet

A printable reference covering coordinate planes, distance formula, horizontal and vertical distances, midpoint formula, and segment lengths for grades 6-8.

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Coordinate geometry connects shapes to numbers by placing points on the coordinate plane. This cheat sheet helps students find distances and midpoints using ordered pairs. These skills are important for graphing, measuring segments, and solving geometry problems with coordinates. A clear reference makes it easier to choose the right formula and avoid sign errors. The main ideas are the coordinate plane, horizontal and vertical distance, the distance formula, and the midpoint formula. Horizontal and vertical distances can be found by subtracting matching coordinates. Diagonal distance uses the Pythagorean theorem in the form $d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$ . The midpoint is found by averaging the $x$ -coordinates and averaging the $y$ -coordinates.

Key Facts

An ordered pair $\left(x, y\right)$ gives a point's horizontal position $x$ and vertical position $y$ on the coordinate plane.
The horizontal distance between $\left(x_1, y\right)$ and $\left(x_2, y\right)$ is $\left|x_2 - x_1\right|$ .
The vertical distance between $\left(x, y_1\right)$ and $\left(x, y_2\right)$ is $\left|y_2 - y_1\right|$ .
The distance between $\left(x_1, y_1\right)$ and $\left(x_2, y_2\right)$ is $d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$ .
The midpoint of a segment with endpoints $\left(x_1, y_1\right)$ and $\left(x_2, y_2\right)$ is $M = \left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right)$ .
Distance is always nonnegative, so a segment length cannot be less than $0$ .
If two points have the same $y$ -coordinate, the segment is horizontal, and if they have the same $x$ -coordinate, the segment is vertical.
For diagonal segments, the changes $x_2 - x_1$ and $y_2 - y_1$ form the legs of a right triangle.

Vocabulary

Coordinate plane: A flat grid formed by the $x$ -axis and $y$ -axis where points are located using ordered pairs.
Ordered pair: A pair of numbers $\left(x, y\right)$ that gives the location of a point on the coordinate plane.
Distance: The length of the segment between two points, found by comparing their coordinates.
Midpoint: The point exactly halfway between two endpoints of a segment.
Endpoint: One of the two points that marks the beginning or end of a segment.
Absolute value: The distance of a number from $0$ on a number line, written as $\left|a\right|$ .

Common Mistakes to Avoid

Subtracting coordinates in the wrong direction without squaring or using absolute value is wrong because distance cannot be negative.
Using the distance formula for horizontal or vertical segments without simplifying first can lead to extra work and sign mistakes.
Averaging only one coordinate for the midpoint is wrong because the midpoint must use both $\frac{x_1 + x_2}{2}$ and $\frac{y_1 + y_2}{2}$ .
Mixing the $x$ -coordinates and $y$ -coordinates is wrong because $x$ measures horizontal change and $y$ measures vertical change.
Forgetting parentheses around negative coordinates is wrong because expressions like $-3 - 5$ and $-3 + 5$ give different results.

Practice Questions

1 Find the distance between $A\left(2, 3\right)$ and $B\left(8, 3\right)$ .
2 Find the midpoint of the segment with endpoints $C\left(-4, 6\right)$ and $D\left(2, -2\right)$ .
3 Find the distance between $E\left(-1, -2\right)$ and $F\left(5, 6\right)$ .
4 Explain why the distance formula is connected to the Pythagorean theorem when two points form a diagonal segment.

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